You seem to have voltage and current conflated.
Voltage is more properly called electromotive force. It does not, in itself, flow, or transfer energy.
Current (usually measured in amperes) is a measure of how much electric charge is moving per unit of time. Current is also not, in itself, a flow of energy.
The flow of energy is called power. To have power, you need both current (\$I\$) and voltage (\$E\$). The power is equal to the product of the two:
$$ P = IE $$
It helps to think about this in terms of analogous mechanical systems, since we can observe mechanical systems directly with our senses. Mechanical systems also have power, where it is equal to the product of force and velocity:
$$ P = Fv $$
If you have force but no velocity, you have no power. An example would be a rubber band stretched between two stationary supports. The band is exerting a force on the supports. This tension is potential energy. But, nothing is moving, and none of that energy stored in the stretched band is being transfered to anything else.
However, if the band can move the supports, now we have velocity. As the band moves the supports, the energy stored in the stretched band will be converted to kinetic energy in the supports. The rate at which this energy transfer happens is power.
Voltage is a force that moves electric charge. Current is the velocity of electric charge. Resistance is how easy it is to move the supports.
Here's a mechanical system that's more analogous to your circuit:
We have a rigid ring, attached to a motor that applies some force to turn it. Also attached to the ring, we have a brake, which resists the turning of the ring. For this analogy to be proper, this has to be a brake that provides a force proportional to the velocity of the ring moving through it. Imagine it's coupled to a fan, so as the ring turns faster, the fan turns faster, creating more aerodynamic drag.
If the motor is applying a force of \$1kN\$, then the brake must be applying an equal force in the opposite direction. If the brake's force is not equal to the motor's, then the ring will experience a net force that will accelerate or decelerate it until the brake's force is equal, and the ring turns at a constant speed. Thus, if the force of the motor is constant, the speed of the ring is a function of the strength of the brake. This is analogous to Ohm's law.
What other forces are acting on the ring? Since we are considering an idealized system with no friction, there are none. If you were to insert strain gauges at points A and B, you would measure a difference between them. B is being compressed as the motor shoves the ring into the brake against its resistance, and A is being stretched as the motor sucks it out of the brake.
But what's the difference between B and C? there is none. If that's not intuitively obvious, consider that you must cut a gap in the ring and insert your hand so this machine can smash it. Is there a point at which you'd prefer to do this? No, your hand will be equally smashed regardless of where you do it on the left side of the ring.
The forces measured by the strain gauges are analogous to voltage. We can only measure voltages relative to some other voltage. That's why your voltmeter has two probes. Wherever you put the black lead is defined as "0V". So, the scenario you present in your question is like measuring the difference between B and C: it is zero.
This seems a little weird, because we know there is a compressive force on that entire side of the ring. It seems like that should be good for something. But consider this: the weight of all the gas in Earth's atmosphere results in a pressure at sea level of about 15 pounds per square inch. Does this mean we can make a machine that's powered just because it's exposed to this pressure? No. In order to do work with this atmospheric pressure, we need a difference in pressure. Without a difference, we can't make the air move. Consider again the definitions of power above, and it should become clear how this is true.
The wire suggested wire is adequate- the resistance is about 130 ohms/km. I got that from kinda a long route. I used the table for copper (AWG) wire resistance per km, adjusted that for the ratio between copper and steel resistivity, and then for the ratio (squared) between the diameters for AWG and the gauge system used for steel wires (whew!) which is "Washburn & Moen; Roebling; or American Steel and Wire".
So only maybe 2.5 ohms for your circuit (based on two wires, 9.1m each, connected at one end.
You might want to use something other than wood for the supports, insulate the wire from the posts, or at least seal the wood with epoxy. If, say, you were in Portland OR, it will (on average) rain 144 days per year and the leakage through the wet wood would tend to drain the battery.
Here are some insulators used on commercial electric fences.
In terms of $ per kWh, AA batteries are a much better deal than 9V batteries, and six AA cells in series would give you 9V. Energy capacity being a lot higher, it also means that you won't need to replace the batteries as often. There is a good reason why 9V batteries are seldom found in modern electronics.
AA cells have a short-circuit current in the amperes, so a current limiting resistor as Andy suggests would be a very good idea, or you could use a blinking LED in series such as this Lumex one, available from distributors such as Digikey for $1 each.
If you see the LED blinking, the wire is shorted.
Best Answer
simulate this circuit – Schematic created using CircuitLab
Consider the two circuits above, we want to measure R1 and R2.
Assume Rw1 to Rw8 represent the wiring resistances.
We know \$ R = \dfrac{V}{I}\$ and all wires have resistance. Assume volt meters are perfect (taking no current) and that the current supplies have internal current meters.
The circuit on the left represents a two wire measurement the current flowing out of R1 causes voltage drops across all three components due to the resistance of the wire. I1 sees the largest voltage, then VM1 then R1. This measurement of R1 is an overestimate because of these drops.
In the circuit on the right we have a 4 wire 'kelvin' connection current flows out of I2 and through R2 so there is slightly more volts across I2 than R2 but we don't care. VM2 is accurately measuring the voltage across R2 because there is no current flowing in that loop so we measure R2 correctly.
In principle these wires could be any length with two obvious limitations
Very long wires from the current source may prevent it from the current required. The current source usually has a low maximum voltage.
The wires to the voltmeter have a high impedance load at one end so may pick up additional signals in an electrically noisy environment. This may be mitigated by using a twisted pair.
I have never seen a problem with cables of the sort of length you typically find in a test lab however.